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A002162 Decimal expansion of the natural logarithm of 2.
(Formerly M4074 N1689)
62
6, 9, 3, 1, 4, 7, 1, 8, 0, 5, 5, 9, 9, 4, 5, 3, 0, 9, 4, 1, 7, 2, 3, 2, 1, 2, 1, 4, 5, 8, 1, 7, 6, 5, 6, 8, 0, 7, 5, 5, 0, 0, 1, 3, 4, 3, 6, 0, 2, 5, 5, 2, 5, 4, 1, 2, 0, 6, 8, 0, 0, 0, 9, 4, 9, 3, 3, 9, 3, 6, 2, 1, 9, 6, 9, 6, 9, 4, 7, 1, 5, 6, 0, 5, 8, 6, 3, 3, 2, 6, 9, 9, 6, 4, 1, 8, 6, 8, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Newton calculated the first 16 terms of this sequence.

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, Sections 1.3.3 and 6.2.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.

Uhler, Horace S.; Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 205-212.

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000

D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications

P. Bala, New series for old functions

Paul Cooijmans, Odds.

X. Gourdon and P. Sebah, The logarithm constant:log(2)

I. Newton, The method of fluxions and infinite series; with its application to the geometry of curve-lines, 1736; see p. 96.

Simon Plouffe, log(2), natural logarithm of 2 to 2000 places

S. Ramanujan, Question 260, J. Ind. Math. Soc.

Eric Weisstein's World of Mathematics, Natural Logarithm of 2, Masser-Gramain Constant, Logarithmic Integral

Wikipedia, Natural logarithm of 2.

FORMULA

log(2) = Sum_{k>=1} 1/(k*2^k) = Sum_{j>=1} (-1)^(j+1)/j.

log(2) = Integral_{t=0..1} dt/(1+t).

log(2) = 2/3 * (1 + sum(k>=1, 2/[(4k)^3-4k] )) (Ramanujan).

log(2) = 4*sum_{k>=0} [3-2*sqrt(2)]^(2k+1)/(2k+1) (Y. Luke). - R. J. Mathar, Jul 13 2006

log(2) = 1-(1/2)*sum_{k>=1} 1/(k*(2k+1)). - Jaume Oliver Lafont, Jan 06 2009, Jan 08 2009

log(2) = 4*sum_{k>=0} 1/((4k+1)(4k+2)(4k+3)). - Jaume Oliver Lafont, Jan 08 2009

log(2) = 7/12+24*sum_{k>=1} 1/(A052787(k+4)*A000079(k)). - R. J. Mathar, Jan 23 2009

From Alexander R. Povolotsky, Jul 04 2009: (Start)

log(2) = 1/4*(3 - sum(n>=1, 1/(n*(n+1)*(2*n+1)) )).

log(2) = (230166911/9240-sum(k>=1, (1/2)^k* (11/k+10/(k+1)+9/(k+2)+8/(k+3)+7/(k+4)+6/(k+5)-6/(k+7)-7/(k+8)-8/(k+9) -9/(k+10)-10/(k+11)) ))/35917. (End)

log(2) = A052882/A000670. - Mats Granvik, Aug 10 2009

From log(1-x-x^2) at x=1/2, log(2)=(1/2)*Sum_{k>=1}L(k)/(k*2^k), where L(n) is the n-th Lucas number (A000032). - Jaume Oliver Lafont, Oct 24 2009

log(2) = Sum_{k>=1} 1/(cos(k*Pi/3)*k*2^k) (Cf. A176900). - Jaume Oliver Lafont, Apr 29 2010

log(2) = (sum(n>=1, 1/(n^2*(n+1)^2*(2*n+1)) ) +11)/16. - Alexander R. Povolotsky, Jan 13 2011

log(2) = sum(n>=1, (2*n+1)/(sum(k=1..n, k^2 )^2)) +396)/576. - Alexander R. Povolotsky, Jan 14 2011

log(2) = 105*(sum(n>=1, 1/(2*n*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)) ) - 319/44100) log(2) = (319/420 - 3/2*sum(n>=1, 1/(6*n^2+39*n+63) )). - Alexander R. Povolotsky, Dec 16 2008

log(2) = sum(k>=1, A191907(2,k)/k ). - Mats Granvik, Jun 19 2011

log(2) = Integral_{x=0..oo} 1/(1 + e^x) dx. - Jean-Fran├žois Alcover, Mar 21 2013

log(2) = limit of zeta(s)*(1-1/2^(s-1)) as s -> 1. - Mats Granvik, Jun 18 2013

From Peter Bala, Dec 10 2013: (Start)

log(2) = 2*sum {n = 1..inf} 1/( n*A008288(n-1,n-1)*A008288(n,n) ), a result due to Burnside.

log(2) = 1/3*sum {n >= 0} (5*n+4)/( (3*n+1)*(3*n+2)*C(3*n,n) )*(1/2)^n = 1/12*sum {n >= 0} (28*n+17)/( (3*n+1)*(3*n+2)*C(3*n,n) )*(-1/4)^n.

log(2) = 3/16*sum {n >= 0} (14*n+11)/( (4*n+1)*(4*n+3)*C(4*n,2*n) )*(1/4)^n = 1/12*sum {n >= 0} (34*n+25)/( (4*n+1)*(4*n+3)*C(4*n,2*n) )*(-1/18)^n. For more series of this type see the Bala link.

See A142979 for series acceleration formulas for log(2) obtained from the Mercator series log(2) = sum {n >= 1} (-1)^(n+1)/n. See A142992 for series for log(2) related to the root lattice C_n. (End)

EXAMPLE

0.693147180559945309417232121458176568075500134360255254120680009493393...

MATHEMATICA

RealDigits[N[Log[2], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

PROG

(PARI) { default(realprecision, 20080); x=10*log(2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b002162.txt", n, " ", d)); } /* Harry J. Smith, Apr 21 2009 */

CROSSREFS

Cf. A016730 Continued fraction. A008288, A142979, A142992.

Sequence in context: A129938 A022698 A013707 * A072365 A239068 A085138

Adjacent sequences:  A002159 A002160 A002161 * A002163 A002164 A002165

KEYWORD

cons,nonn

AUTHOR

N. J. A. Sloane, Apr 30 1991

STATUS

approved

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Last modified July 26 01:07 EDT 2014. Contains 244923 sequences.